Next we will obtain regularity of \(\phi \)  in time. First by global Schauder estimates (see [1]}), there is a \(C=C(m,\nabla m,R,d)\)  such that
\( \sup _{(x,t) \in B_R(0) \times [0,T]} |\phi (x,t)| + \sup _{(x,t) \in B_R(0) \times [0,T]} \sum _{i=1}^{d} |\partial _{x_i} \phi (x,t)| + \sup _{(x,t) \in B_R(0) \times [0,T]} \sum _{i,j=1}^{d} |\partial ^2_{x_ix_j} \phi (x,t)| \le C.  \) 
